# Joined Common Chords of Napoleon's Circumcircles

### What is this about?

### An Equilateral Triangle in Napoleon's Circumcircles

Let $ABC',$ $BCA',$ and $CAB'$ be Napoleon's triangles constructed on the sides of $\Delta ABC.$ Choose point $D$ on the circumcircle $ABC',$ pass it through $A$ to the intersection $F$ with $C(CAB')$ and through $E$ to the intersection $E$ with $C(BCA').$

Triangles $DEF$ is equilateral.

### Hint

The problem is very simple; it submits to chasing inscribed angles.

### Solution

The solution is outlined in an old variant of this problem.

### Acknowledgment

The problem described on this page stems from an observation of Hirotaka Ebisui.

### Napoleon's Theorem

- Napoleon's Theorem
- A proof with complex numbers
- A second proof with complex numbers
- A third proof with complex numbers
- Napoleon's Theorem, Two Simple Proofs
- Napoleon's Theorem via Inscribed Angles
- A Generalization
- Douglas' Generalization
- Napoleon's Propeller
- Napoleon's Theorem by Plane Tessellation
- Fermat's point
- Kiepert's theorem
- Lean Napoleon's Triangles
- Napoleon's Theorem by Transformation
- Napoleon's Theorem via Two Rotations
- Napoleon on Hinges
- Napoleon on Hinges in GeoGebra
- Napoleon's Relatives
- Napoleon-Barlotti Theorem
- Some Properties of Napoleon's Configuration
- Fermat Points and Concurrent Euler Lines I
- Fermat Points and Concurrent Euler Lines II
- Escher's Theorem
- Circle Chains on Napoleon Triangles
- Napoleon's Theorem by Vectors and Trigonometry
- An Extra Triple of Equilateral Triangles for Napoleon
- Joined Common Chords of Napoleon's Circumcircles
- Napoleon's Hexagon
- Fermat's Hexagon
- Lighthouse at Fermat Points
- Midpoint Reciprocity in Napoleon's Configuration
- Another Equilateral Triangle in Napoleon's Configuration
- Yet Another Analytic Proof of Napoleon's Theorem
- Leo Giugiuc's Proof of Napoleon's Theorem
- Gregoie Nicollier's Proof of Napoleon's Theorem
- Fermat Point Several Times Over

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